In a medium bowl, combine the ground pork, apple, onion, bread crumbs, sage, salt, and pepper. Mix until well combined. Form into 4 good sized patties. You can probably make 8 thinner patties if you wanted to, but I'm a fan of a big juicy burger.
Preheat oven to 375 F.
Heat a large oven proof skillet under medium heat and add olive oil. Brown the patties on each side, about 2-3 minutes per side. You don't want to cook them all the way through, just get a nice crust on the outside.
Place skillet in oven and cook for about 20 minutes, or until cooked through.
Meanwhile, you can fry your eggs. I did mine one at a time and I like them over-easy. You can cook them up any way you wish, but make sure you have a nice oozy (but still done) yolk!
Open Kaiser rolls and butter both sides. Place in oven for about 5 minutes to toast.
To assemble burger, place patty on bottom bun, top with egg, then cheese and cover with top bun. In my opinion, this doesn't need any condiment. The oozing egg is just what it needs and the burger is one of the juiciest burgers I've ever eaten.